If ⁻¹ 1 3 + ⁻¹ 3 5 + ⁻¹ x= 2 then x=

Step-by-step solution

Given: ⁻¹ 1 3 + ⁻¹ 3 5 + ⁻¹ x = 2 Rewriting the equation: ⁻¹ 1 3 + ⁻¹ 3 5 = 2 - ⁻¹ x Using the identity 2 - ⁻¹ x = ⁻¹ x , we obtain: ⁻¹ 1 3 + ⁻¹ 3 5 = ⁻¹ x Let = ⁻¹ 1 3 and = ⁻¹ 3 5 , so that = 1 3 and = 3 5 . Since , [0, 2 ] , both angles are in the first quadrant and their cosines are positive: = 1 - ( 1 3 )^2 = 2 2 3 = 1 - ( 3 5 )^2 = 4 5 The equation becomes + = ⁻¹ x , so taking cosine of both sides: ( + ) = x Applying the cosine addition formula: x = - = ( 2 2 3 ) ( 4 5 ) - ( 1 3 ) ( 3 5 ) Simplifying: x = 8 2